Algebraic Expressions and Identities  Class 8  Mathematics
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Exercise 6.1  Algebraic Expressions and Identities  R.D. Sharma  Mathematics  Class 8
Identify the terms, their coefficients for each of the following expressions:
(i) $7 x^{2} y z5 x y$
(ii) $x^{2}+x+1$
(iii) $3 x^{2} y^{2}5 x^{2} y^{2} z^{2}+z^{2}$
(iv) $9a b+b cc a$
(v) $\frac{a}{2}+\frac{b}{2}a b$
(vi) $0.2 x0.3 x y+0.5 y$
For each expression, we'll identify the terms and then the coefficients for each term:
(i) Expression: $7 x^2 y z  5 x y$
Terms: $7 x^2 y z$, $5 x y$
Coefficients: $7$ for $x^2 y z$, $5$ for $x y$
(ii) Expression: $x^2 + x + 1$
Terms: $x^2$, $x$, $1$
Coefficients: $1$ for $x^2$, $1$ for $x$, $1$ for the constant term
(iii) Expression: $3 x^2 y^2  5 x^2 y^2 z^2 + z^2$
Terms: $3 x^2 y^2$, $5 x^2 y^2 z^2$, $z^2$
Coefficients: $3$ for $x^2 y^2$, $5$ for $x^2 y^2 z^2$, $1$ for $z^2$
(iv) Expression: $9  a b + b c  c a$
Terms: $9$, $a b$, $b c$, $c a$
Coefficients: $1$ for the constant term $9$, $1$ for $a b$, $1$ for $b c$, $1$ for $c a$
(v) Expression: $\frac{a}{2} + \frac{b}{2}  a b$
Terms: $\frac{a}{2}$, $\frac{b}{2}$, $a b$
Coefficients: $\frac{1}{2}$ for $a$, $\frac{1}{2}$ for $b$, $1$ for $ab$
(vi) Expression: $0.2 x  0.3 x y + 0.5 y$
Terms: $0.2 x$, $0.3 x y$, $0.5 y$
Coefficients: $0.2$ for $x$, $0.3$ for $x y$, $0.5$ for $y$
Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any category?
(i) $x+y$
(ii) 1000
(iv) $7+a+5 b$
(v) $2 b3 b^{2}$
(vii) $5 x4 y+3 x$
(viii) $4 a15 a^{2}$
(x) $p q r$
(xi) $p^{2} q+p q^{2}$
(iii) $x+x^{2}+x^{3}+x^{4}$
(vi) $2 y3 y^{2}+4 y^{3}$
(ix) $x y+y z+z t+t x$
(xii) $2 p+2 q$
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Sign up nowExercise 6.2  Algebraic Expressions and Identities  R.D. Sharma  Mathematics  Class 8
Add the following algebraic expressions:
(i) $3 a^{2} b,4 a^{2} b, 9 a^{2} b$
(ii) $\frac{2}{3} a, \frac{3}{5} a,\frac{6}{5} a$
(iii) $4 x y^{2}7 x^{2} y, 12 x^{2} y6 x y^{2},3 x^{2} y+5 x y^{2}$
(iv) $\frac{3}{2} a\frac{5}{4} b+\frac{2}{5} c, \frac{2}{3} a\frac{7}{2} b+\frac{7}{2} c, \frac{5}{3} a+\frac{5}{2} b\frac{5}{4} c$
(v) $\frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x,\frac{11}{2} y\frac{12}{5} x\frac{13}{7} x y$
(vi) $\frac{7}{2} x^{3}\frac{1}{2} x^{2}+\frac{5}{3}, \frac{3}{2} x^{3}+\frac{7}{4} x^{2}x+\frac{1}{3}, \frac{3}{2} x^{2}\frac{5}{2} x2$
The results for the addition of the given algebraic expressions are:
(i) $8 a^{2} b$
(ii) $\frac{a}{15}$
(iii) $2 x^2 y + 3 x y^2$
(iv) $\frac{23}{6} a  \frac{9}{4} b + \frac{53}{20} c$
(v) $\frac{51}{14} x y  \frac{19}{35} x  \frac{31}{10} y$
(vi) $5 x^3 + \frac{11}{4} x^2  \frac{7}{2} x$
These results are simplified forms of the expressions after addition.
Subtract:
(i) $5 x y$ from $12 x y$
(ii) $2 a^{2}$ from $7 a^{2}$
(iii) $2 ab$ from $3 a5 b$
(iv) $2 x^{3}4 x^{2}+3 x+5$ from $4 x^{3}+x^{2}+x+6$
(v) $\frac{2}{3} y^{3}\frac{2}{7} y^{2}5$ from $\frac{1}{3} y^{3}+\frac{5}{7} y^{2}+y2$
(vi) $\frac{3}{2} x\frac{5}{4} y\frac{7}{2} z$ from $\frac{2}{3} x+\frac{3}{2} y\frac{4}{3} z$
(vii) $x^{2} y\frac{4}{5} x y^{2}+\frac{4}{3} x y$ from $\frac{2}{3} x^{2} y+\frac{3}{2} x y^{2}\frac{1}{3} x y$
(viii) $\frac{a b}{7}\frac{35}{3} b c+\frac{6}{5} a c$ from $\frac{3}{5} b c\frac{4}{5} a c$
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Sign up nowTake away:
(i) $\frac{6}{5} x^{2}\frac{4}{5} x^{3}+\frac{5}{6}+\frac{3}{2} x$ from $\frac{x^{3}}{3}\frac{5}{2} x^{2}+\frac{3}{5} x+\frac{1}{4}$
(ii) $\frac{5 a^{2}}{2}+\frac{3 a^{3}}{2}+\frac{a}{3}\frac{6}{5}$ from $\frac{1}{3} a^{3}\frac{3}{4} a^{2}\frac{5}{2}$
(iii) $\frac{7}{4} x^{3}+\frac{3}{5} x^{2}+\frac{1}{2} x+\frac{9}{2}$ from $\frac{7}{2}\frac{x}{3}\frac{x^{2}}{5}$
(iv) $\frac{y^{3}}{3}+\frac{7}{3} y^{2}+\frac{1}{2} y+\frac{1}{2}$ from $\frac{1}{3}\frac{5}{3} y^{2}$
(v) $\frac{2}{3} a c\frac{5}{7} a b+\frac{2}{3} b c$ from $\frac{3}{2} a b\frac{7}{4} a c\frac{5}{6} b c$
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Sign up nowSubtract $3 x4 y7 z$ from the sum of $x3 y+2 z$ and $4 x+9 y11 z$.
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Sign up nowSubtract the sum of $3 l4 m7 n^{2}$ and $2 l+3 m4 n^{2}$ from the sum of $9 l+2 m3 n^{2}$ and $3 l+m+4 n^{2} \ldots .$.
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Sign up nowSubtract the sum of $2 xx^{2}+5$ and $4 x3+7 x^{2}$ from 5 .
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Sign up nowSimplify each of the following:
(i) $x^{2}3 x+5\frac{1}{2}\left(3 x^{2}5 x+7\right)$
(ii) $[53 x+2 y(2 xy)](3 x7 y+9)$
(iii) $\frac{11}{2} x^{2} y\frac{9}{4} x y^{2}+\frac{1}{4} x y\frac{1}{14} y^{2} x+\frac{1}{15} y x^{2}+\frac{1}{2} x y$
(iv) $\left(\frac{1}{3} y^{2}\frac{4}{7} y+11\right)\left(\frac{1}{7} y3+2 y^{2}\right)\left(\frac{2}{7} y\frac{2}{3} y^{2}+2\right)$
(v) $\frac{1}{2} a^{2} b^{2} c+\frac{1}{3} a b^{2} c\frac{1}{4} a b c^{2}\frac{1}{5} c b^{2} a^{2}+\frac{1}{6} c b^{2} a\frac{1}{7} c^{2} a b+\frac{1}{8} c a^{2} b$.
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Sign up nowExtra Questions  Algebraic Expressions and Identities  R.D. Sharma  Mathematics  Class 8
Select the statements that are true.
A $(x) y = x(y)$
B $(x) y = (x y)$
C $x(y) = (x y)$
D $x(y) = (x y)$
Selected true statements:
A: $(x)y = x(y)$
B: $(x)y = (xy)$
C: $x(y) = (xy)$
Explanation:
The multiplication involving negatives can be understood by considering how signs interact with each other:

The expression $(x)y$ implies multiplying a negative value $x$ with a positive value $y$. According to sign multiplication rules, a negative times a positive is negative, thus $(x)y = (xy)$.

Similarly, $x(y)$ translates to multiplying a positive value $x$ with a negative value $y$. Following the same rule, a positive times a negative yields a negative, leading to $x(y) = (xy)$.

Both expressions $(x)y$ and $x(y)$ essentially end up implying the same arithmetic operation: multiply $x$ and $y$ and then invert the sign. Therefore, we deduce $(x)y = x(y)$.
This reasoning demonstrates that statements A, B, and C are true, while statement D is false, as it incorrectly equates $x(y)$ to $(xy)$, neglecting the sign change.
Find the value of $(997)^{2}$ using algebraic identities.
A) 994009
B) 996009
C) 940009
D) 946009
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